If you don't have a large wooden bowl, the garlic and salt can be mashed into a paste on a cutting board using the side of a chef's knife.
Preheat oven to 400 degrees.
Submerge romaine leaves in a large bowl of ice water for 10 minutes. Then thoroughly dry in a salad spinner or colander, and place in a single layer on a paper towel-lined dish cloth. Roll up cloth; refrigerate until lettuce is crisp and completely dry (any water clinging to leaves may cause the dressing to break), at least 2 hours and up to 2 days.
Place garlic in a large wooden bowl, add 1/4 teaspoon salt, and mash to a paste with a fork. Transfer half of paste to a rimmed baking sheet; set bowl aside. With the fork, stir butter and 2 tablespoons oil into garlic paste on sheet. Toss with bread cubes to evenly coat. Sprinkle with 1/4 cup cheese, season with salt and pepper, and toss again to evenly coat. Spread into a single layer. Bake, stirring halfway through, until crisp and golden, 12 to 15 minutes. Let cool to room temperature on sheet.
Add anchovies to bowl with garlic paste and mash to a paste with a fork. Whisk in lemon juice, Worcestershire, Dijon, and egg yolk. Season with 1/4 teaspoon each salt and pepper. Slowly add remaining 1/3 cup oil in a thin stream, whisking constantly. Add lettuce and toss to evenly coat, then toss with remaining 3/4 cup cheese and croutons. Serve immediately.